Introduction to Modeling II: Questions and Assumptions

This is the second post in a series on engineering modeling. In the first post, Introduction to Modeling I: Overview, I showed you a simple example system, a basic pendulum, that will be used to talk about some modeling concepts.

Before we start constructing any model, we need to create a list of questions the model is supposed to answer. In this pendulum example, we might want to know things like:

• if the rod will be strong enough to support the weight
• how the pendulum will move in a variety of conditions
• what kind of force would be required to get the pendulum to move in a certain way

Here we will look at a model that addresses the first question. We will explore the other questions in later posts.

Much of the coursework engineers go through in college is dedicated to learning how to create models for a variety of systems that predict behavior to answer questions. One interesting activity engineers engage in is gathering knowledge about a system, whether from past school work, research publications, commercial software, or other resources, and integrating it all together into a model that answers important questions about a system, which in turn helps engineers make decisions. To answer our first question about the pendulum, we will look at material that engineering students might learn in classes on statics (the study of forces in things that are not moving) and solid mechanics (the study of what happens to solid objects when you apply force to them).

Every model is an approximation of a real system. Approximate models are based on assumptions about a system and its environment; if these assumptions were all completely true, then the model would be 100% accurate. In reality, assumptions are only partially correct. Adding more assumptions can simplify a model, but can also make it less accurate. An engineer must manage the tradeoff between model accuracy and simplicity. Many assumptions are reasonable to make, but engineers need to be careful or they might get very unexpected results from the real system. Have a look at what happened when bridge engineers assumed that vibrations caused by wind blowing across a bridge had no effect:

Let’s start off with a very simple model for our pendulum, and assume the following:

• The weight has a mass of m = 5 kilograms
• The 2mm thick rod is made of an aluminum alloy with a yield stress (explained below) of 20 MPa
• The rod is much less massive than the weight at the bottom, so we can neglect the mass of the rod
• The aluminum is homogeneous, that is, it has the same properties everywhere inside the rod. It has no spots in the rod that are weaker than others.
• The pendulum is not moving

A solid object can break apart when the stress inside gets too high. You can think of stress as a type of internal pressure; it has the same units: force per area (PSI in U.S. customary units), just like pressure in a liquid or gas. In metric the unit for pressure is a Pascal (Pa), which is defined to be one Newton per square meter. The symbol normally used to represent a stress value is sigma (). Stress is a little more complicated concept than pressure in a fluid. Stress can be positive (compressive stress) or negative (tensile stress), and the direction of the stress matters. In the case of the pendulum, we will focus on just one type of stress: axial stress (), stress that occurs due to a force along the length of an object. The drawing below illustrates the idea of axial stress.

Because of gravity pulling down on the weight with a force of Newtons (where is the acceleration of gravity), there is an internal axial force along the length of the rod of 49.05 N. In the drawing above I show an imaginary cut through the rod. Imagine yourself being in the middle of the cut. To keep the weight from falling, you would have to pull the two halfs of the rod together with a force of 49.05 N. Each cross section of the rod must also resist of force of 49.05 N. The cross-sectional area of the rod is , where , and is the diameter of the rod, which is 2 millimeters (mm), or 0.002 meters (m). This square meters of aluminum must resist 49.05 N of force without breaking at every cross-section of the rod.

Let’s think conceptually about axial stress in a rod. If the force on the rod goes up, then the stress inside goes up proportionately. If we want to reduce the stress, we can make the rod thicker. This line of thinking is reflected in a very simple model for axial stress: , where is the tension in the rod. We can rewrite this equation, our model for stress, in terms of the rod diameter and the mass at the bottom of the pendulum: . We can see from this equation that increasing mass increases axial stress, while increasing the rod diameter reduces stress.

Different materials have different tolerance for stress. We say that a material that can handle more stress than another is stronger. This material property can be quantified using something called yield stress. A material will yield, or stretch past the point in can return back to its normal shape, when the stress inside exceeds its yield stress. A material will break when the stress reaches an even higher level, called the ultimate strength or rupture stress of the material. We are assuming that the rod here is made of a type of aluminum that will yield when the stress exceed 20 MPa (mega-Pascals: one mega-Pascal is one million Pascals). We can determine the yield stress and rupture stress of a material using a machine like the one in the video below (something happens at 53 seconds):

By plugging numbers for mass and diameter into our equation for stress, we can calculate that the axial stress in the rod (when the pendulum is not moving) is 15.6 MPa, which is less than the yield stress of 20 MPa, so our model predicts that the rod will in fact be strong enough to support the weight. Congratulations! We have answered the first question using an engineering model.

When might our model for whether the rod will be strong enough break down? What conditions can you think of that would cause some of the assumptions not to hold? What things does the model not account for?

Posted: May 15th, 2009 | Filed under: Modeling |

---